14.8 Lagrange Multipliers: Optimizing with a Constraint Some optimization problems involve finding the extreme values of a function f(x,y) subject to a constraint g(x,y)=o We are not seeking the minimum value of f(x,y) (which is 0), but rather the minimum among all points (x, y) that lie on the line A point P = (a,b) satisfying these equations is called a critical point Theorem 1: for the optimization problem with constraint and f(a,b) is called a critical value Assume that f(x,y) and g(x,y) are The method of Lagrange multipliers is valid in any number of variables differentiable functions. if f(x,y) has a The method of Lagrange multipliers can be used when there is more local minimum or a local maximum on the than one constraint equation, but we must add another multiplier for constraint curve g(x,y) = 0 at p = (a,b), each additional constraint then: rfp = ¸rgp